Abstract
We present homogenization results for stochastic semilinear parabolic equations in varying domains which are stochastic counterparts of the some fundamental results of Khruslov and Marchenko, Skrypnik, Cioranescu, Dal Maso and Murat.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Allaire, Grégoire Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), no. 6, 1482–1518.
Bensoussan, A.: Homogenization of a class of stochastic partial differential equations, In Composite Media and Homogenization Theory, Birkhauser, Basel and Boston, MA, 1991, pp. 47–65.
Bensoussan, A.; Lions, J.L.; Papanicolaou, G.C.: Asymptotic analysis for periodic structures, North-Holland, Amsterdam, 1978.
Bensoussan A.: Some existence results for stochastic partial differential equations, in Stochastic Partial Differential Equations and Applications (Trento, 1990), Pitman Res. Notes Math. Ser. 268, Longman Scientific and Technical, Harlow, UK, 1992, pp. 37–53.
Bourgeat, A.; Mikelić, A.; Wright, S.: Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math. 456 (1994), 19–51.
Buttazzo, Giuseppe; Dal Maso, Gianni Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23 (1991), no. 1, 17–49.
Casado Diaz, J., Gayte, I.: The two-scale convergence method applied to generalized Besicovitch spaces. Proc. R. Soc. Lond. A 458, 2925–2946 (2002).
Cioranescu, D.; Damlamian, A.; Griso, G. The periodic unfolding method in homogenization. In “Assyr, Banasiak, Damlamian, Sango: Multiple scales problems in biomathematics, mechanics, physics and numerics”, 1–35, GAKUTO Internat. Ser. Math. Sci. Appl., 31, Gakkō tosho, Tokyo, 2009.
Cioranescu, Doina; Donato, Patrizia An introduction to homogenization. Oxford Lecture Series in Mathematics and its Applications, 17. The Clarendon Press, Oxford University Press, New York, 1999.
Cioranescu, D; Donato, P.; Murat, F.; Zuazua, E.: Homogenization and corrector for the wave equation in domains with small holes, Ann. Scuola Norm. Sup. Pisa 18 (1991), 251–293.
Cioranescu, Doina; Murat, François A strange term coming from nowhere [MR0652509; MR0670272]. Topics in the mathematical modelling of composite materials, 45–93, Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser Boston, Boston, MA, 1997.
Da Prato G., Zabczyk J.: Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992.
Dal Maso, Gianni; Mosco, Umberto Wiener’s criterion and Γ-convergence. Appl. Math. Optim. 15 (1987), no. 1, 15–63
Dal Maso, G.; Murat, F.: Asymptotic behaviour and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24 (1997), no. 2, 239–290.
Dal Maso, Gianni An introduction to Γ -convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA, 1993. xiv+340 pp.
L.C. Evans, Partial Differential Equations : Graduate text in Maths, AMS, Providence, 1998.
L.C. Evans, R.F. Gariepy : Measure Theory and Fine Properties of Functions, CRS Press, 1992.
Gilbarg, David; Trudinger, Neil S.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001.
Ichihara, Naoyuki.: Homogenization for stochastic partial differential equations derived from nonlinear filterings with feedback. J. Math. Soc. Japan 57 (2005), no. 2, 593–603.
Ichihara, Naoyuki.: Homogenization problem for stochastic partial differential equations of Zakai type. Stoch. Stoch. Rep. 76 (2004), no. 3, 243–266.
Kozlov, S. M. The averaging of random operators. (Russian) Mat. Sb. (N.S.) 109(151) (1979), no. 2, 188–202, 327.
Marchenko, V.A.; Khruslov, E.Ya.: Boundary value problems in regions with fine grained boundaries, Naukova Dumka, Kiev, 1974.
Marchenko, V. A.; Khruslov, E. Ya.: Homogenization of partial differential equations. Progress in Mathematical Physics, 46. Birkh äuser, Boston, 2006.
Mohammed, M.; Sango, M., Homogenization of linear hyperbolic stochastic partial differential equation with rapidly oscillating coefficients: the two scale convergence method. Asymptot. Anal. 91 (2015), no. 3–4, 341–371.
Mohammed, M.; Sango, M.,Homogenization of Neumann problem for hyperbolic stochastic partial differential equations in perforated domains. Asymptot. Anal. 97 (2016), no. 3–4, 301–327.
Mohammed, M,; Homogenization of nonlinear hyperbolic stochastic equation via Tartar’s method. J. Hyperbolic Differ. Equ. 14 (2017), no. 2, 323–340.
Nagase, N.: Remarks on nonlinear stochastic partial differential equations : an application of the splitting-up method, SIAM J. Control & Optim., 33(1995), 1716–1730.
Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), no. 3, 608–623.
Nguetseng, G., Homogenization structures and applications I, Z. Anal. Anwend 22 (2003) 73–107.
Nguetseng, Gabriel; Sango, Mamadou; Woukeng, Jean Louis Reiterated ergodic algebras and applications. Comm. Math. Phys. 300 (2010), no. 3, 835–876.
Papanicolaou, G.C.; Varadhan, S.R.S: Diffusions in regions with many small holes, Stochastic Differential Systems : Filtering and Control, Lecture Notes in Control and Information Sciences, Vol. 25, Springer Verlag, Berlin, …, 1980, pp190–206.
Papanicolaou, G.C.; Varadhan S.R.S.: Boundary value problems with rapidly oscillating random coefficients, Proceedings of a colloquium on random fields : rigorous results in Statistical Mechanics and Quantum Field Theory, Colloq. Math. Soc. J. Bolyai, North-Holland, Amsterdam, 1979.
Prokhorov, Yu. V.: Convergence of random processes and limit theorems in probability theory. Teor. Veroyatnost. i Primenen. 1 (1956), 177–238.
Razafimandimby, Paul André; Woukeng, Jean Louis Homogenization of nonlinear stochastic partial differential equations in a general ergodic environment. Stoch. Anal. Appl. 31 (2013), no. 5, 755–784.
Razafimandimby, Paul André; Sango, Mamadou; Woukeng, Jean Louis Homogenization of a stochastic nonlinear reaction-diffusion equation with a large reaction term: the almost periodic framework. J. Math. Anal. Appl. 394 (2012), no. 1, 186–212.
Sango, M.: Asymptotic behavior of a stochastic evolution problem in a varying domain. Stochastic Anal. Appl. 20 (2002), no. 6, 1331–1358.
Sango, M.: Homogenization of the Dirichlet problem for a system of quasilinear elliptic equations in a domain with fine-grained boundary. Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003), no. 2, 183–212.
Sango, M.: Pointwise a priori estimates for solution of a system of quasilinear elliptic equation. Appl. Anal. 80 (2001), no. 3–4, 367–378.
Sango, M.: Homogenization of stochastic semilinear parabolic equations with non-Lipschitz forcings in domains with fine grained boundaries. Commun. Math. Sci. 12 (2014), no. 2, 345–382.
Sango, M.; Woukeng, J-L.: Stochastic sigma-convergence and applications, Dynamics of Partial Differential Equations, 8 (2011), 4, 261–310.
Simon, J.: Compact sets in the space \(L^{p}\left (0,T;B\right ) \), Annali Mat. Pura Appl., 1987, 146, IV, 65–96.
Skorokhod, A. V.: Limit theorems for stochastic processes. Teor. Veroyatnost. i Primenen. 1 (1956), 289–319.
Skrypnik I.V.: Methods for analysis of nonlinear elliptic boundary value problems, Nauka, Moscow, 1990. English translation in : Translations of Mathematical Monographs, 139, AMS, Providence, 1994.
Skrypnik, I. V.: Averaging of quasilinear parabolic problems in domains with fine-grained boundaries. (Russian) Differentsial’nye Uravneniya 31 (1995), no. 2, 350–363, 368; translation in Differential Equations 31 (1995), no. 2, 327–339.
Wang, W.; Cao, D.; Duan, J.: Effective macroscopic dynamics of stochastic partial differential equations in perforated domains. SIAM J. Math. Anal. 38 (2006/07), no. 5, 1508–1527
Wang, W.; Duan, J.: Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions. Comm. Math. Phys. 275 (2007), no. 1, 163–186.
Zhikov, V. V.; Kozlov, S.M.; Oleinik, O. A.: Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994.
Zhikov, V.V., Krivenko, E.V.: Homogenization of singularly perturbed elliptic operators. Matem. Zametki. 33, 571–582 (1983)
Acknowledgements
The research of the authors is supported by the National Research Foundation of South Africa under the grant CGRR 93459. The support of DFG and AIMS for the participation of Mamadou Sango to the AIMS-DFG workshop in Mbour, Senegal is gratefully appreciated.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this chapter
Cite this chapter
Mohammed, M.A.Y., Sango, M. (2018). Homogenization of Stochastic Parabolic Equations in Varying Domains. In: Schulz, V., Seck, D. (eds) Shape Optimization, Homogenization and Optimal Control . International Series of Numerical Mathematics, vol 169. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90469-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-90469-6_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-90468-9
Online ISBN: 978-3-319-90469-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)